Simultaneous convexification of bilinear functions over polytopes with application to network interdiction

Danial Davarnia, Jean Philippe P. Richard, Mohit Tawarmalani

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

We study the simultaneous convexification of graphs of bilinear functions g k (x; y) = yA k x over x ∈ Ξ = {x ∈ [0, 1] n |Ex ≥ f} and y ∈ ∆m = {y ∈ R m + |1y ≤ 1}. We propose a constructive procedure to obtain a linear description of the convex hull of the resulting set. This procedure can be applied to derive convex and concave envelopes of certain bilinear functions, to study unary expansions of integer variables in mixed integer bilinear sets, and to obtain convex hulls of sets with complementarity constraints. Exploiting the structure of Ξ, the procedure naturally yields stronger linearizations for bilinear terms in a variety of practical settings. In particular, we demonstrate the effectiveness of the approach by strengthening the traditional dual formulation of network interdiction problems and report encouraging preliminary numerical results.

Original languageEnglish (US)
Pages (from-to)1801-1833
Number of pages33
JournalSIAM Journal on Optimization
Volume27
Issue number3
DOIs
StatePublished - Jan 1 2017
Externally publishedYes

Keywords

  • Bilinear functions
  • Convex hulls
  • Cutting planes
  • Envelopes
  • Network interdiction

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